H∞ direct adaptive fuzzy control with unknown control gain for uncertain nonlinear systems

With unknown control gain function and plant bound functions, a H^∞ direct adaptive fuzzy controller (AFC) for a class of single-input single-output (SISO) uncertain affine nonlinear systems under external disturbance is proposed. A conventional fuzzy logic system (FLS) is used to approximate the control gain function, and a FLS with variable universes of discourse, which has the characteristic of high fuzzy output precision with only one adjusting parameter, is used to approximate a ideal controller. A novel fuzzy approximation theorem is introduced to solve the problem of unknown control gain function in direct AFC scheme. Both adaptive laws of conventional FLS and variable universe FLS are derived by virtue of the Lyapunov stability theorem. Under the assumption that the total approximation error is bounded, it is proved that the closed-loop system not only is stable in sense that all variables are bounded, but also achieves the H∞ tracking performance and the tracking error convergence. Simulation example is demonstrated to confirm the effectiveness of this approach.